On the structure of linear phase perfect reconstruction quincunx filter banks

The paper is motivated by the desire to extend the notion of Quillen-Suslin (Lam 1978) type result on the unimodular completion of polynomial matrices to the context of multidimensional finite impulse response (FIR) linear phase (LP) perfect reconstruction (PR) filter bank design. The specific problem is to construct a multidimensional linear phase PR filter bank, given a subset, say, one of the filters of the filter bank. In filter bank terminology, given one of the filters satisfying the linear phase property (and the property that its analysis polyphase components do not have any finite common zero set) the problem is to complete the linear phase filter bank. If linear phase property is not desired, the construction of the PR filter bank follows almost trivially via an application of the Quillen-Suslin theorem. However, since the linear phase property of the filter bank dictates certain symmetries, the matrix completion problem becomes somewhat more constrained. We only deal with the case of the so called quincunx sub-sampling in the two-dimensional (2D) case, and show that in this simple case, Quillen-Suslin type result indeed holds true with the additional linear phase constraint. While the result was previously announced in (Basu 1998) proofs have not appeared in the open literature.